Question

A pair of tangents is drawn to a unit circle with centre at the origin and these tangents intersect at $A$ enclosing an angle of ${60}^{\circ }.$ The area enclosed by these tangents and the arc of the circle is

• A. $\frac{2}{\sqrt{3}}-\frac{\pi }{6}$ sq. units
• B. $\sqrt{3}-\frac{\pi }{3}$ sq. units
• C. $\frac{\pi }{3}-\frac{\sqrt{3}}{6}$ sq. units
• D. $\sqrt{3}(1-\frac{\pi }{6})$ sq. units

Answer 1

The correct option is B $\sqrt{3}-\frac{\pi }{3}$ sq. units

$r=1$
Let length of tangent be $L$
Now the quadrilateral formed by the tangents and radii is cyclic quadrilateral
$\therefore \angle A+\angle O=180°$
$\Rightarrow \tan {60}^{\circ }=\frac{L}{r}$
$\therefore L=\sqrt{3}$

Area of shaded region $=L\cdot r-\frac{1}{2}{r}^2\theta =\sqrt{3}-\frac{\pi }{3}$ sq. units